Clusters, spherical particles

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

When micelles are formed just above the cmc, they are spherical aggregates in which surfactant molecules are clustered, tails together, to form a spherical particle. At higher concentrations the amount of excess surfactant is such that the micelles acquire a rod shape or, eventually, even a layer structure. 

The UPS indicated structure change is associated with size reduction as the discontinuous gold film is transformed into rod-shape and spherical particles with size of 5-10 nm. Accordingly, with size reduction the activity displayed in CO oxidation is also altered the rate increased from 6.7 X 10 to 2 X 10 molmin cm . Consequently, not only the gold-reducible oxide interaction is responsible for the increased activity, but also size reduction. Indeed, small clusters themselves are able to activate the reaction components shown by theoretical calculations performed for 10-15-atom clusters, which can activate easily oxygen [177,200], but in real catalyst, even at the smallest active ensemble, it consists of a few hundreds atoms. 

Any study of colloidal crystals requires the preparation of monodisperse colloidal particles that are uniform in size, shape, composition, and surface properties. Monodisperse spherical colloids of various sizes, composition, and surface properties have been prepared via numerous synthetic strategies [67]. However, the direct preparation of crystal phases from spherical particles usually leads to a rather limited set of close-packed structures (hexagonal close packed, face-centered cubic, or body-centered cubic structures). Relatively few studies exist on the preparation of monodisperse nonspherical colloids. In general, direct synthetic methods are restricted to particles with simple shapes such as rods, spheroids, or plates [68]. An alternative route for the preparation of uniform particles with a more complex structure might consist of the formation of discrete uniform aggregates of self-organized spherical particles. The use of colloidal clusters with a given number of particles, with controlled shape and dimension, could lead to colloidal crystals with unusual symmetries [69]. 

From both the mean sizes and the appearance characteristics it seems that at lower concentrations of CuCl2 solution, normally spherical particles of uniform sizes are produced while for CCuci2 = 0.4 kmol-m 3 needle-like crystalline clusters possibly up to several micrometers long are formed, which consists of the growing particles in the... 

Fig. 4.20 DEM-simulated packing density under gravity for a two-dimensional spherical particulate assembly 50 pm in diameter, with f =fw = 0.364 and with van der Waals forces, showing the formation of clusters, which decrease the packing density by more than 10%. [Reprinted by permission from Y. F. Cheng, S. J. Guo, and H. Y. Li, DEM Simulation of Random Packing of Spherical Particles, Powder Technol., 107, 123-130 (2000).]...

Whereas the surface area of a crystalline silica is in fact the external surface area, the surface area and the pore size distribution of an amorphous silica are actually determined by the dimensions of the silica spheres (primary particles) that build up the network. For non-aggregated spherical particles, this relationship is very straightforward. In this silica type, the primary particles are not clustered and Sheinfain s3 globular theory can be applied. The globular theory predicts an inverse relationship between surface area and the primary particle size by the following equation ... 

Clusters consist of from five to several hundred spherical particles attached to each other in a similar fashion to a bunch of grapes. They occur infrequently and appear to be a product of high power or high velocity. Flakes of smokeless powder are few in number and are occasionally seen in promptly collected residue. Unlike the other three types of particles, which are inorganic in nature, the flakes are organic although sometimes spherical particles are embedded in their surface. They range in size from about 50 to 1,000 pm. Clusters and powder flakes are rarely seen in casework as they are... 

Finally, we direct attention to Barnes et al. (1987), for their extensive review of applications of computer simulations to dense suspension rheology, and also to Hassonjee et al. (1988), for their numerical scheme for dealing with large clusters of spherical particles. 

Studies on the rheology of two-phase particulate systems suggest the existence of a deformation threshold that depends on the concentration of particles in the composite. Below this threshold (—68 vol.% for spherical particles), deformation occurs primarily by the flow of the composite matrix. Particles increase the effective viscosity of the matrix by absorbing energy and by forming clusters. 

Effective Conductivity of Dilute Mixtures, The simplest, best defined case, is a cluster of spherical particles dispersed in a liquid and located in a... 

The AEM study has shown that, for loading around 10 RUO2 molecules percm (formally equivalent to about 10 monolayers), the oxide phase is essentially organized in cluster of particles of approximately spherical shape. The size of the clusters may reach values of 50 to 60 nm. For lower loadings no particles can be detected, possibly because of their quite small size. 

The purpose of calculating Henry s Law constants is usually to determine the parameters of the adsorption potential. This was the approach in Ref. [17], where the Henry s Law constant was calculated for a spherically symmetric model of CH4 molecules in a model microporous (specific surface area ca. 800 m /g) silica gel. The porous structure of this silica was taken to be the interstitial space between spherical particles (diameter ca. 2.7 nm ) arranged in two different ways as an equilibrium system that had the structure of a hard sphere fluid, and as a cluster consisting of spheres in contact. The atomic structure of the silica spheres was also modeled in two ways as a continuous medium (CM) and as an amorphous oxide (AO). The CM model considered each microsphere of silica gel to be a continuous density of oxide ions. The interaction of an adsorbed atom with such a sphere was then calculated by integration over the volume of the sphere. The CM model was also employed in Refs. [36] where an analytic expression for the atom - microsphere potential was obtained. In Ref. [37], the Henry s Law constants for spherically symmetric atoms in the CM model of silica gel were calculated for different temperatures and compared with the experimental data for Ar and CH4. This made it possible to determine the well-depth parameter of the LJ-potential e for the adsorbed atom - oxygen ion. This proved to be 339 K for CH4 and 305 K for Ar [37]. On the other hand, the summation over ions in the more realistic AO model yielded efk = 184A" for the CH4 - oxide ion LJ-potential [17]. Thus, the value of e for the CH4 - oxide ion interaction for a continuous model of the adsorbent is 1.8 times larger than for the atomic model. 

The discussions in the previous sections of this chapter have focused on the thermodynamics of single particles. However, there is an important class of problems involving the. statistical properties of interacting clouds of particles in the molecular cluster size range. The size distribution of these particles can be calculated using a simple spherical particle model as... 

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